Find all functions $f:\mathbb R \to \mathbb R$ such that \[ xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2, \] for all real numbers $x,y$. [i]Proposed by B.J. Venkatachala[/i]

Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

Pick $x, y, z$ to be real numbers satisfying $(-x+y+z)^2-\frac13 = 4(y-z)^2$, $(x-y+z)^2-\frac14 = 4(z-x)2$, and $(x+y-z)^2 -\frac15 = 4(x-y)^2$. If the value of $xy+yz +zx$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$. [i]Proposed by Ali Zamani, Hooman Fattahi[/i]

Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times? [i]Proposed by Matthew Weiss

For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$

Let $p, q, r, s, t \in \mathbb{R}^{*}_{+}$ satisfying: i) $p^2 + pq + q^2 = s^2$ ii) $q^2 + qr + r^2 = t^2$ iii) $r^2 + rp + p^2 = s^2 - st + t^2$ Prove that \[\frac{s^2 - st + t^2}{s^2t^2} = \frac{r^2}{q^2t^2} + \frac{p^2}{q^2s^2} - \frac{pr}{q^2ts}\]

The maximum value of \[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \] over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.

The incenter of triangle $ABC$ is $ I$. the circumcircle of $ABC$ is tangent to $BC$, $CA$, $AB$ at $T, E, F$. $R$ is a point on $BC$ . Let the $C$-excenter of $\vartriangle CER$ be $L$. Prove that points $L,T,F$ are collinear if and only if $B,E,A,R$ are concyclic. [i]proposed by kyou46[/i]

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$, define an [i]$A$-inversion[/i] of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds: [list] [*]$a_i \ge w_i > w_j$ [*]$w_j > a_i \ge w_i$, or [*]$w_i > w_j > a_i$. [/list] Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$, and for any positive integer $k$, the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$-inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$-inversions.

The tangent lines to the circumcircle of triangle ABC passing through vertices $B$ and $C$ intersect at point $F$. Points $M$, $L$ and $N$ are the feet of the perpendiculars from vertex $A$ to the lines $FB$, $FC$ and $BC$ respectively. Show that $AM+AL \geq 2AN$

In the quadrilateral $ABCD$ the angles $B{}$ and $D{}$ are straight. The lines $AB{}$ and $DC{}$ intersect at $E$ and the lines $AD$ and $BC$ intersect at $F{}.$ The line passing through $B{}$ parallel to $C{}$D intersects the circumscribed circle $\omega$ of $ABF{}$ at $K{}$ and the segment $KE{}$ intersects $\omega$ at $P{}.$ Prove that the line $AP$ divides the segment $CE$ in half.

Let $ABCD$ be a trapezoid such that $|AC|=8$, $|BD|=6$, and $AD \parallel BC$. Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|PS|=5$, find the area of the trapezoid $ABCD$.

Let $ t_{k} \left(n\right)$ show the sum of $ k^{th}$ power of digits of positive number $ n$. For which $ k$, the condition that $ t_{k} \left(n\right)$ is a multiple of 3 does not imply the condition that $ n$ is a multiple of 3? $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$

Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is $\textbf{(A)}\ 43\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 47$

Let $x, y$ be positive real numbers with $x + y + xy= 3$. Prove that$$x + y\ge 2.$$ When does equality holds? (K. Czakler, GRG 21, Vienna)

A segment $AB$ is given and it's midpoint $K$. On the perpendicular line to $AB$, passing through $K$ a point $C$, different from $K$ is chosen. Let $N$ be the intersection of $AC$ and the line passing through $B$ and the midpoint of $CK$. Let $U$ be the intersection point of $AB$ and the line passing through $C$ and $L$, the midpoint of $BN$. Prove that the ratio of the areas of the triangles $CNL$ and $BUL$, is independent of the choice of the point $C$.

Let $a_1, a_2, \ldots, a_n$ be positive reals for $n \geq 2$. For a permutation $(b_1, b_2, \ldots, b_n)$ of $(a_1, a_2, \ldots, a_n)$, define its $\textit{score}$ to be $$\sum_{i=1}^{n-1}\frac{b_i^2}{b_{i+1}}.$$ Show that some two permutations of $(a_1, a_2, \ldots, a_n)$ have scores that differ by at most $3|a_1-a_n|$.

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

Let $A$ be a square matrix of odd order (at least $3$) whose entries are odd integers. Prove that if $A$ is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.

Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying \[ f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z). \] Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0$ or $2.$

Ding and Jianing are playing a game. In this game, they use pieces of paper with $2014$ positions, in which some permutation of the numbers $1, 2, \dots, 2014$ are to be written. (Each number will be written exactly once). Ding fills in a piece of paper first. How many pieces of paper must Jianing fill in to ensure that at least one of her pieces of paper will have a permutation that has the same number as Ding’s in at least one position?

Given the binary operation $\ast$ defined by $a\ast b=a^b$ for all positive numbers $a$ and $b$. The for all positive $a,b,c,n,$ we have $\textbf{(A) }a\ast b=b\ast a\qquad\textbf{(B) }a\ast (b\ast c)=(a\ast b)\ast c\qquad$ $\textbf{(C) }(a\ast b^n)=(a\ast n)\ast b\qquad\textbf{(D) }(a\ast b)^n=a\ast (bn)\qquad \textbf{(E) }\text{None of these}$

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]